One of the staple events from GSC is the Bug Catching Contest. The Bug Catching Contest is a special contest that takes place on Tuesdays, Thursdays & Saturdays and has you sent out into the National Park to capture Bug Pokémon.

You get sent in with 20 Park Balls and you have to capture the best Pokémon. However, unlike the Safari Zone, you get to choose one of your Pokémon to go in with and fight the wild Pokemon with.
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Capturing Pokémon

You can only keep one Pokémon from the Bug Catching Contest on you at any one time. With that in mind, you have to consider all factors after capturing a second Pokémon. Once a second Pokémon is captured, you get prompted whether or not you wish to keep it or to replace the previous Pokémon.

Things to consider are the Pokémon's held item, level, stats, and HP at time of capture. When you replace the previously caught Pokémon, you will no longer have it at all so don't take the decision lightly. Your best bet to win the contest is to catch a Pinsir or Scyther.

Prizes

Once you have either caught a Pokémon you think will win or have run out of Park Balls, you can go to the exit and decide to opt out of the rest of the contest. Otherwise, you will have to wait the full twenty minutes or have your Pokemon knocked out to exit. You will be taken to the east entrance of the National Park and prizes will be given based on your Pokémon. The top prize is very rare so it's best to always try to enter and win.